Non-destructive estimation of particle size using X-ray diffraction technique full re
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SYNOPSIS

Non- destructive estimation of particle size of finished products is essentially required to judge their suitability for intended applications. For this estimation, a ceramic heating element (SiC rod) of a heat Treatment furnace at 1400 Degree celcius has been chosen. The particle size has been estimated with the proposed method based on counting reflection spots in a back reflection X-Ray pattern. Calibration curve has been drawn on the basis of the data obtained by standard sieve method as well as proposed X-Ray method from three powder samples of the same material with different particle sizes. The particle size estimated by this proposed method is found to closely match to that by sieve method.

INTRODUCTION
Nondestructive testing is the use of tests to examine a material, to detect imperfections, determine its properties or assess its quality without changing its usefulness. Nondestructive testing is term generally applied to certain tests that are used in industrial operations. These techniques are used in the field of medicine as well. Industrial uses of nondestructive testing include monitoring the quality of products in a manufacturing process. It is also useful in the maintenance of process machinery. Nondestructive testing is commonly employed in ship building, in manufacturing of aerospace vehicles and automobiles, metals manufacturing, railroad maintenance, electric plant construction and maintenance.



X-RAY DIFFRACTION TECHNIQUE
Modern microscopy is beginning to show as actual crystal structure directly, many useful techniques have been developed in which crystals are photographed by process involving the absorption or diffraction of x-rays, electrons or neutrons.

X-rays are of electromagnetic in nature and behave as light and it has a short wavelength. Its wavelength ranges from 0.3 to 0.07 nano meter, depending on soft medium hard x-rays. In the case of visible light grating with the seperation between 1000 and 2000 nano meter are used to diffract wavelengths in range from 400-to 800-nano meter. In crystal, however the seperation between equally spaced parallel rows of atoms or order of a few nanometer.
Diffraction studies are used to analysis crystalline structures when x-rays of a given frequency strike as atom or ion they interact with its electrons causing them to vibrate with the frequency of x-ray beam since the electrons become vibrating electrical changes they radiate the x-rays with no changes in frequency. These reflected rays come off the atom or ions in any direction. In other words the electrons of an atom scatter the x-ray beam in all directions. This is the process of diffraction and analysis of the patterns of diffracted radiations is usually based upon the Bragg equation.
BRAGGâ„¢S LAW:







In the above figure two parallel lines AA and BB represent places of atoms that reflect incident x-rays. The waves may be reflected from an atom at M1 and M2 . However x-rays are not only reflected from the surfaces plane and also from subsurface plane. The path difference for rays L2M1N and L2M2N2 reflected from adjacent planes is the length PM2Q which is equal to 2dSinO where d is the atomic or the inter planar spacing of a given plane and O the angle of incidence of the radiation. If this path difference is equal to an integral number of incidences of the radiation if this path difference is equal to an integral number of wavelength the reflected rays from adjacent plane will be in phase and will reinforce one another only in certain specific directions. When depend on the angle of the incident x-rays as well as the spacing between atoms in the crystal. The relationship is expressed in the Bragg equation
2dSin = n
n = Order of diffraction
n = 1,2,3¦¦.


EXPERIMENT:
THEORY:
Ceraramics are used in various important products, e.g. refractory bricks, grinding wheels, heating elements, ceramic wares, etc. The particle size of the ceramics has a marked effect on their strength which is inversely proportional to the square root of the particle size. The knowledge of particle size of the ceramic products becomes quite often necessary for their utility. Sieve method for the determination of particle size is a widely accepted standard method. However, this method is not at all applicable for finished solid products. Therefore, alternative method of non-destructive estimation of particle size is necessary.
The effect of particle size on the X-ray diffraction pattern is well known and it can be utilized for the determination of particle size. Particles of the order of 100 µm give rise to spotty diffraction pattern. As the particle size decreases, the diffraction spots become smaller in size and greater in number ultimately coalescing into continuous lines in a Debye-Scherrer pattern when size of the particle is of the order of 10 µm. Hence, the particle size of 10 µm and above can be estimated and the distribution of sizes can be visualized by recording a transmission or back reflection pattern and either by counting the number of spots or by measuring the diameter of the spots. As the particle sizes of the ceramics products are usually bigger than 10 µm, the method of counting reflection spots is most suitable.
EXPERIMENTAL
DEVELOPMENT OF THE METHOD:
If a specimen, with particle sizes bigger than 10 µm, is irradiated with a fine pencil of monochromatic X-rays and a back reflection pattern is recorded, then a number of spotty rings are observed in the film. The number of spots (N), can be related with the mean particle size (P) using expression:

P= [(V.P. cos.d)/2N]1/3 ¦¦¦¦(1)


Where, V = volume irradiated, p = multiplicity factor, = Bragg angle, 1/2cos = probability that a particular particle will reflect and d = divergence of the beam.
The irradiated volume (V), cannot be known directly. Since it depends on the maximum depth to which the X-ray beam can penetrate and then re-emerge from the surface to produce a visible record on the film. This difficulty is overcome by substituting a parameter ˜C™ in place of [(Vp cos d)/2]1/3. This parameter ˜C™ depends only on the particle sizes of the material provided that the pattern of a particular material with different particle sizes are recorded under identical conditions and the number of spots are counted for the same ring. For a particular material, the variation of ˜C™ with ˜P™ may be considered as linear within a short span of P, say 40 µm.The value of the parameter C is to be determined by recording a number of patterns of the same material in the powder form and with different particle sizes (Ps) within a particular range. Ps is to determine by Sieve method and the corresponding number of spots (N) are counted. The value of C can be computed from equation (1). From the C versus Ps curve, the value of C (viz.CM) corresponding to the mid-point of the range of selected Ps is chosen for the minimization of estimation error. Then, by X-ray method, any particle size Px within this range can be evaluated from the expression:
Px = CM/N 1/3 ¦¦¦¦..(2)
RESULTS
CALIBRATION:
A ceramic heating element of a heat treatment furnace of capacity 1400° c is chosen for the non-destructive estimation of its particle size. For identification of the material, a Debye-Scherrer pattern is recorded in a 57.3 mm dia camera using CoKa radiation. The observed visual intensities (VI) and the calculated ˜d™ values of the diffraction lines are given in Table1.These values are found to compare favorably with the published values 4 for SiC, which are also incorporated in Table1. Hence, the material of the heating element is identified as SiC.
Table-1 Visual Intensity(VI) and Ëœdâ„¢ values of diffraction lines


No of lines Visual Intensity (VI)
d() Published Ëœdâ„¢ values of SiC (A)
1 W 2.617 2.62
2 VVS 2.515 2.50
3 MS 2.364 2.35
4 W 2.180 2.17
5 VW 2.005 1.99
6 VW 1.678 1.675
7 VS 1.542 1.540
8 MS 1.421 1.420
9 VS 1.315 1.312
10 W 1.294 1.290
11 VW 1.261 1.257
12 VVW -------- 1.220
13 VVW -------- 1.134
14 W 1.092 1.090
15 W 1.044 1.044
16 VVW ------- 1.005
17 MS 1.000 1.000
18 W 0.990 0.989
19 MS 0.976 0.975
20 VVW ------- 0.956
21 MS 0.944 0.941
22 VW 0.916 0.912

V = very, S = strong, W = weak, M= medium

Table 2 “ Counted number of spots (N) and calculated values of C, CM and
Px of the powder samples with particle sizes Ps.
Sample No. Ps (µm) N C CM Px (µm)
1 63 75 .0265 - 59.52
2 45 175 .0251 .0251 45.00
3 32 395 .0235 - 34.21

The back reflection pattern (fig.1-3) of the three samples are, then, recorded in a flat camera using CoKa radiation, the specimen to film distance being 3.0 cm. All the patterns show a set of spotty rings. The number of spots in a ring is found to increase and the size of the spots is found to decrease as the particle size decreases. The number of spots (N) in the second ring of 4.5 cm dia, is counted by placing the X-ray plates on a sensitive film comparator and observing the spots through a magnifying glass at a magnification of 6x. From the values of N and Ps, the values of C are calculated (Table 2).




Fig.4 shows mean linear curve of C versus Ps from where the value of CM is determined, as reported in Table2.

Fig.4 “ Variation of C with particle size.
Using this CM, the particle sizes Px (by X-ray method), are calculated by using the expression (2) and substituting the corresponding values of N (as presented in Table 2). The Px versus Ps curve (fig.5) is found to be a straight line and is utilized here, as the calibration curve for the non-destructive estimation of particle size.



The back reflection pattern (fig.6) of the actual sample of heating element (SiC rod) is recorded under the condition identical with those for the powder samples of SiC used for calibration. On examination of the pattern under a film comparator, it is observed that in the second ring of 4.5 cm dia, the individual spots do not crowd together or overlap which indicates that the result has practical significance. The number of spots in this ring is counted in a similar manner as earlier and the particle size Px is calculated by the expression (2). The corresponding particle size (Ps) is estimated from the calibration curve.
The estimated values of particle size of the SiC rod are: N=240, CM=0.0251; Px=40.4µm and Ps=39.5 µm; it is found that the particle size estimated by the developed method closely corresponds to that by the Sieve method.


Fig. 6 “ Back reflection pattern of SiC rod

DISCUSSION:
The X-ray method for estimation of particle size of finished products is very simple, requiring the counting of the number of spots in the diffraction ring of back reflection pattern. The only condition is that a calibration is to be drawn for which at least three samples of the same material in powder form with different sizes are required.
For determining the value of the parameter Cn an average straight line C versus Ps curve has been drawn though the plotted points are not collinear. This can be justified for a particular range of short span. This range is to be selected from the expected particle size of the product to be tested.
For counting the number of spots (N) accurately, a particular Debye-Scherrer ring is to be chosen in which the individual spots do not crowd together or overlap.
The method has technological importance as it can be applied to finished products with out destroying any of their properties. It also takes in to account the occurrence and distribution of flat, elongated and other shapes of particles, which have an important bearing on the properties of engineering materials.
CONCLUSION:
The particle size of the finished SiC rod used us heating element of high temperature heat treatment furnace is non-destructively estimated from the recorded back reflection X-ray pattern where the standard Sieve method is not applicable. This estimation essential to ensure the use of such products for practical purposes.
ADVANTAGES:
It does not affect the affect the entire structure or it does not alter the usefulness of the material cheaper than the ultra sonic method.
DISADVANTAGES:
It requires more calibration steps. So calculation is difficult. For finding the size, different particle sizes are needed.

REFERENCES





SYNOPSIS

Non- destructive estimation of particle size of finished products is essentially required to judge their suitability for intended applications. For this estimation, a ceramic heating element (SiC rod) of a heat Treatment furnace at 1400 Degree celcius has been chosen. The particle size has been estimated with the proposed method based on counting reflection spots in a back reflection X-Ray pattern. Calibration curve has been drawn on the basis of the data obtained by standard sieve method as well as proposed X-Ray method from three powder samples of the same material with different particle sizes. The particle size estimated by this proposed method is found to closely match to that by sieve method.

INTRODUCTION
Nondestructive testing is the use of tests to examine a material, to detect imperfections, determine its properties or assess its quality without changing its usefulness. Nondestructive testing is term generally applied to certain tests that are used in industrial operations. These techniques are used in the field of medicine as well. Industrial uses of nondestructive testing include monitoring the quality of products in a manufacturing process. It is also useful in the maintenance of process machinery. Nondestructive testing is commonly employed in ship building, in manufacturing of aerospace vehicles and automobiles, metals manufacturing, railroad maintenance, electric plant construction and maintenance.



X-RAY DIFFRACTION TECHNIQUE
Modern microscopy is beginning to show as actual crystal structure directly, many useful techniques have been developed in which crystals are photographed by process involving the absorption or diffraction of x-rays, electrons or neutrons.

X-rays are of electromagnetic in nature and behave as light and it has a short wavelength. Its wavelength ranges from 0.3 to 0.07 nano meter, depending on soft medium hard x-rays. In the case of visible light grating with the seperation between 1000 and 2000 nano meter are used to diffract wavelengths in range from 400-to 800-nano meter. In crystal, however the seperation between equally spaced parallel rows of atoms or order of a few nanometer.
Diffraction studies are used to analysis crystalline structures when x-rays of a given frequency strike as atom or ion they interact with its electrons causing them to vibrate with the frequency of x-ray beam since the electrons become vibrating electrical changes they radiate the x-rays with no changes in frequency. These reflected rays come off the atom or ions in any direction. In other words the electrons of an atom scatter the x-ray beam in all directions. This is the process of diffraction and analysis of the patterns of diffracted radiations is usually based upon the Bragg equation.
BRAGGâ„¢S LAW:





In the above figure two parallel lines AA and BB represent places of atoms that reflect incident x-rays. The waves may be reflected from an atom at M1 and M2 . However x-rays are not only reflected from the surfaces plane and also from subsurface plane. The path difference for rays L2M1N and L2M2N2 reflected from adjacent planes is the length PM2Q which is equal to 2dSinO where d is the atomic or the inter planar spacing of a given plane and O the angle of incidence of the radiation. If this path difference is equal to an integral number of incidences of the radiation if this path difference is equal to an integral number of wavelength the reflected rays from adjacent plane will be in phase and will reinforce one another only in certain specific directions. When depend on the angle of the incident x-rays as well as the spacing between atoms in the crystal. The relationship is expressed in the Bragg equation
2dSin = n
n = Order of diffraction
n = 1,2,3¦¦.

EXPERIMENT:
THEORY:
Ceraramics are used in various important products, e.g. refractory bricks, grinding wheels, heating elements, ceramic wares, etc. The particle size of the ceramics has a marked effect on their strength which is inversely proportional to the square root of the particle size. The knowledge of particle size of the ceramic products becomes quite often necessary for their utility. Sieve method for the determination of particle size is a widely accepted standard method. However, this method is not at all applicable for finished solid products. Therefore, alternative method of non-destructive estimation of particle size is necessary.
The effect of particle size on the X-ray diffraction pattern is well known and it can be utilized for the determination of particle size. Particles of the order of 100 µm give rise to spotty diffraction pattern. As the particle size decreases, the diffraction spots become smaller in size and greater in number ultimately coalescing into continuous lines in a Debye-Scherrer pattern when size of the particle is of the order of 10 µm. Hence, the particle size of 10 µm and above can be estimated and the distribution of sizes can be visualized by recording a transmission or back reflection pattern and either by counting the number of spots or by measuring the diameter of the spots. As the particle sizes of the ceramics products are usually bigger than 10 µm, the method of counting reflection spots is most suitable.
EXPERIMENTAL
DEVELOPMENT OF THE METHOD:
If a specimen, with particle sizes bigger than 10 µm, is irradiated with a fine pencil of monochromatic X-rays and a back reflection pattern is recorded, then a number of spotty rings are observed in the film. The number of spots (N), can be related with the mean particle size (P) using expression:

P= [(V.P. cos.d)/2N]1/3 ¦¦¦¦(1)

Where, V = volume irradiated, p = multiplicity factor, = Bragg angle, 1/2cos = probability that a particular particle will reflect and d = divergence of the beam.
The irradiated volume (V), cannot be known directly. Since it depends on the maximum depth to which the X-ray beam can penetrate and then re-emerge from the surface to produce a visible record on the film. This difficulty is overcome by substituting a parameter ˜C™ in place of [(Vp cos d)/2]1/3. This parameter ˜C™ depends only on the particle sizes of the material provided that the pattern of a particular material with different particle sizes are recorded under identical conditions and the number of spots are counted for the same ring. For a particular material, the variation of ˜C™ with ˜P™ may be considered as linear within a short span of P, say 40 µm.The value of the parameter C is to be determined by recording a number of patterns of the same material in the powder form and with different particle sizes (Ps) within a particular range. Ps is to determine by Sieve method and the corresponding number of spots (N) are counted. The value of C can be computed from equation (1). From the C versus Ps curve, the value of C (viz.CM) corresponding to the mid-point of the range of selected Ps is chosen for the minimization of estimation error. Then, by X-ray method, any particle size Px within this range can be evaluated from the expression:
Px = CM/N 1/3 ¦¦¦¦..(2)
RESULTS
CALIBRATION:
A ceramic heating element of a heat treatment furnace of capacity 1400° c is chosen for the non-destructive estimation of its particle size. For identification of the material, a Debye-Scherrer pattern is recorded in a 57.3 mm dia camera using CoKa radiation. The observed visual intensities (VI) and the calculated ˜d™ values of the diffraction lines are given in Table1.These values are found to compare favorably with the published values 4 for SiC, which are also incorporated in Table1. Hence, the material of the heating element is identified as SiC.
Table-1 Visual Intensity(VI) and Ëœdâ„¢ values of diffraction lines
No of lines Visual Intensity (VI)
d() Published Ëœdâ„¢ values of SiC (A)
1 W 2.617 2.62
2 VVS 2.515 2.50
3 MS 2.364 2.35
4 W 2.180 2.17
5 VW 2.005 1.99
6 VW 1.678 1.675
7 VS 1.542 1.540
8 MS 1.421 1.420
9 VS 1.315 1.312
10 W 1.294 1.290
11 VW 1.261 1.257
12 VVW -------- 1.220
13 VVW -------- 1.134
14 W 1.092 1.090
15 W 1.044 1.044
16 VVW ------- 1.005
17 MS 1.000 1.000
18 W 0.990 0.989
19 MS 0.976 0.975
20 VVW ------- 0.956
21 MS 0.944 0.941
22 VW 0.916 0.912
V = very, S = strong, W = weak, M= medium
Table 2 “ Counted number of spots (N) and calculated values of C, CM and
Px of the powder samples with particle sizes Ps.
Sample No. Ps (µm) N C CM Px (µm)
1 63 75 .0265 - 59.52
2 45 175 .0251 .0251 45.00
3 32 395 .0235 - 34.21

The back reflection pattern (fig.1-3) of the three samples are, then, recorded in a flat camera using CoKa radiation, the specimen to film distance being 3.0 cm. All the patterns show a set of spotty rings. The number of spots in a ring is found to increase and the size of the spots is found to decrease as the particle size decreases. The number of spots (N) in the second ring of 4.5 cm dia, is counted by placing the X-ray plates on a sensitive film comparator and observing the spots through a magnifying glass at a magnification of 6x. From the values of N and Ps, the values of C are calculated (Table 2).


Fig.4 shows mean linear curve of C versus Ps from where the value of CM is determined, as reported in Table2.

Fig.4 “ Variation of C with particle size.
Using this CM, the particle sizes Px (by X-ray method), are calculated by using the expression (2) and substituting the corresponding values of N (as presented in Table 2). The Px versus Ps curve (fig.5) is found to be a straight line and is utilized here, as the calibration curve for the non-destructive estimation of particle size.


The back reflection pattern (fig.6) of the actual sample of heating element (SiC rod) is recorded under the condition identical with those for the powder samples of SiC used for calibration. On examination of the pattern under a film comparator, it is observed that in the second ring of 4.5 cm dia, the individual spots do not crowd together or overlap which indicates that the result has practical significance. The number of spots in this ring is counted in a similar manner as earlier and the particle size Px is calculated by the expression (2). The corresponding particle size (Ps) is estimated from the calibration curve.
The estimated values of particle size of the SiC rod are: N=240, CM=0.0251; Px=40.4µm and Ps=39.5 µm; it is found that the particle size estimated by the developed method closely corresponds to that by the Sieve method.


Fig. 6 “ Back reflection pattern of SiC rod

DISCUSSION:
The X-ray method for estimation of particle size of finished products is very simple, requiring the counting of the number of spots in the diffraction ring of back reflection pattern. The only condition is that a calibration is to be drawn for which at least three samples of the same material in powder form with different sizes are required.
For determining the value of the parameter Cn an average straight line C versus Ps curve has been drawn though the plotted points are not collinear. This can be justified for a particular range of short span. This range is to be selected from the expected particle size of the product to be tested.
For counting the number of spots (N) accurately, a particular Debye-Scherrer ring is to be chosen in which the individual spots do not crowd together or overlap.
The method has technological importance as it can be applied to finished products with out destroying any of their properties. It also takes in to account the occurrence and distribution of flat, elongated and other shapes of particles, which have an important bearing on the properties of engineering materials.
CONCLUSION:
The particle size of the finished SiC rod used us heating element of high temperature heat treatment furnace is non-destructively estimated from the recorded back reflection X-ray pattern where the standard Sieve method is not applicable. This estimation essential to ensure the use of such products for practical purposes.
ADVANTAGES:
It does not affect the affect the entire structure or it does not alter the usefulness of the material cheaper than the ultra sonic method.
DISADVANTAGES:
It requires more calibration steps. So calculation is difficult. For finding the size, different particle sizes are needed.

REFERENCES
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